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Let $\mathbb{k}$ be an algebraically closed field of positive characteristic, $X$ an affine smooth variety over it. Then the ring of crystalline differential operators on $X$ is generated by $\mathcal{O}(X)$ and $\operatorname{Der}_\mathbb{k} \mathcal{O}(X)$ with relations $f.\partial=f\partial$, $\partial.f-f.\partial=\partial(f)$, $\partial.\partial'-\partial'.\partial=[\partial,\partial']$, $f \in \mathcal{O}(X), \partial,\partial' \in \operatorname{Der}_\mathbb{k} \mathcal{O}(X)$. An important example are the Weyl algebras over $\mathbb{k}$.

I'm looking for papers that discuss ring theoretical aspects of this ring. I know that it is a left and right Noetherian domain, finitely generated, and the description of its center, but nothing more.

I am also interested in the question of when this notion of differential operators is better suited than the classical Grothendieck's notion of differential operator.

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    $\begingroup$ As you note the ring of crystalline differential operators are noetherian, whereas Grothendieck's ring won't be in positive characteristic. So that's already a big advantage. $\endgroup$
    – Donu Arapura
    Commented Apr 4, 2023 at 23:07

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Let $\pi: T^\ast X \to X$ be the projection of the cotangent bundle and $-'$ denote Frobenius twist. Then $D(X)$ is an Azumaya algebra over its center $\pi_\ast \mathcal O_{T^*X'}$. This appears in Roman Bezrukavnikov, Ivan Mirković, Dmitriy Rumynin: "Localization of modules for a semisimple Lie algebra in prime characteristic."

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